The expanded group of tie knots was inspired by more complicated knots, like the Trinity knot and the Elderidge knot, that have more elaborate facades, and are tied using the thinner end of a tie.

The process of tying a tie is essentially just a series of moves. The tie wearer's shirt is broken up into three regions — left, right, and center. At each step, the blade of the tie starts in one of those regions and moves into one of the other regions, alternating between going over and under the existing knot.

At certain points, it is also possible to tuck the blade of the tie under a fold of cloth in the knot. For conventional tie knots, this is only done at the end to secure the knot. For the expanded universe of knots developed in the paper, tucks are possible throughout the process, leading to more complex textures.

The mathematicians, led by Mikael Vejdemo-Johansson of the KTH Royal Institute of Technology in Stockholm, made a rigorous formal language to describe the moves involved in tying a knot. The letter "T" represents a "turnwise" move — taking the tie blade and moving it to the region clockwise in the mirror of its current region. If the tie end is in the center region, near the wearer's neck, a turnwise move sends it to the right. If the end is on the right, it goes over or under the knot to the left.

"W" means "widdershins" — a counterclockwise move going in the opposite direction. These moves can be seen in the image below, taken from the paper:

The formal alphabet is completed with "U" is for tucking the tie "under" some previous bow, or fold, in the knot.

With this alphabet in place, a tie knot can be described by a sequence of these symbols. The authors of the paper introduce a few rules for making knots — because of practical considerations, only sequences of up to eleven T and W winding moves are allowed. They also observe that a tuck needs to have two moves in the same direction (either TT or WW) immediately before it, so that there is a fold of cloth to tuck the blade into.

Page 2 of 2 - One of the advantages of using these notations to describe the knot-tying process is that it lends itself well to mathematical analysis. Counting sequences like this is a fairly straightforward process. The authors found 2046 valid sequences of T and W winding moves, and by including optional tucks, a total of 177,147 possible ways to knot a tie.

Another advantage is that it is possible for a computer program to actually write out all of these tie knots. The authors made such a program, in the form of a random tie knot generator that includes a knot sequence chosen from the 177,147 possibilities, along with pictures to help guide the process.

The majority of these knots, like the Trinity and Elderidge knots that inspired them, are tied using the thin blade as the active blade. Many of the knots look pretty strange.

We took a few of the random knots from the generator and tried them out: